Number

number is a primitive mathematical entity, whose concept arises from the need to count, as an abstraction of the concept of quantity, or to assign the position in a list of elements, or to identify the relationship between quantities of the same type.

Numerical sets (number systems)

Numbers can be classified into sets, called number systems:

  • Natural numbers (counting numbers) \(\mathbb{N}\)
  • Whole numbers
  • Integers \(\mathbb{Z}\)
  • Rational numbers \(\mathbb{Q}\)
  • Real numbers \(\mathbb{R}\)
  • Complex numbers \(\mathbb{C}\)

Natural numbers (counting numbers)

The most basic numbers used in algebra are those we use to count objects: \(1, 2, 3, 4, 5, …\) and so on. These are called the counting numbers also called natural numbers.

The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Natural numbers are an ordered succession, that is, they have an order in such a way as to establish (in ascending order) that each natural number that precedes another is less than the next (and obviously each number that follows is greater than the previous one). Therefore, for each natural number (excluding zero) always exists a previous number and the next one.

In practice 1 is less than 2, just as 2 is greater than 1 (so on for all numbers). The following mathematical symbols are used to indicate these relationships (and others):

  • \(<\) less-than
  • \(>\) greater-than
  • \(\leq\) less than or equal to
  • \(\geq\) greater than or equal to
  • \(=\) equal to
  • \(\neq\) not equal to
  • \(\ll\) much less than
  • \(\gg\) much greater than

Whole numbers

Including zero with the counting numbers gives a new set of numbers called the whole numbers.

\[0, 1, 2, 3, 4, 5 …\]

The discovery of the number zero was a big step in the history of mathematics.

Integers

The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign).

Rational numbers

rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator.

Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.

Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them.

Real numbers

real number is a value of a continuous quantity that can represent a distance along a line, they include all the measuring numbers. Every real number corresponds to a point on the number line.

Complex numbers

complex number is a number that can be expressed in the form \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is a solution of the equation \(x^2 = −1\).

Because no real number satisfies this equation, \(i\) is called an imaginary number. For the complex number \(a + bi\), \(a\) is called the real part, and \(b\) is called the imaginary part.

Parity (odd and even numbers)

An odd number is an integer that is not divisible by 2 (without remainder). Odd numbers are: \(\pm 1, \pm 3, \pm 5, \pm 7, \pm 9, \pm 11, …\)

An integer that is not odd is called even number. An even number is an integer divisible (without remainder) by 2. Even numbers are: \(0, \pm 2, \pm 4, \pm 6, \pm 8, \pm 10, …\)

An integer \(k\) is even if and only if the last digit of \(k\) is even. For example, 2139856 is even, while 2146903 is not even (but odd). An integer not divisible by 2 (i.e. leaving remainder 1) is called an odd number.

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